Variable Phase and Frequency Pulse-Width Modulation Technique

ABSTRACT

Phased array systems rely on the production of an exact carrier frequency to function. Reconstructing digital signals by specified amplitude and phase is accomplished explicitly by inducing frequency shifts away from a base frequency implied by phase changes. Shifting the carrier frequency of a digitally controlled phased array while preserving the timing of the individual phase pulses enables more efficient driving of the phased array system when the phase of the drive signals change dynamically in time.

RELATED APPLICATION

This application claims the benefit of U.S. Provisional PatentApplication No. 62/744,656, filed on Oct. 12, 2018, which isincorporated by reference in its entirety.

FIELD OF THE DISCLOSURE

The present disclosure relates generally to reconstructing digitalsignals by specified amplitude and phase via inducing frequency shiftsaway from a base frequency implied by phase changes.

BACKGROUND

Phased array systems rely on the production of an exact carrierfrequency to function. To simplify systems, it is often assumed that thecarrier frequency is emitted during all relevant times so that thesystem can be treated as time invariant. This time invariance isnecessary for the input signals to the array element transducers to betreated as complex values.

Generating a constant frequency pulse-width modulated (PWM) digitalsignal with a given phase offset for all relevant times is trivial. Butchanging the state of a phased array system often involves changing thephase angle of the elements, which violates the time-invariancerequirement. This results in many side-effects, including a shift infrequency. Since the digital signal generation assumes that the basefrequency (the frequency with which the primitive phase angles arespecified relative to) is equal to the carrier frequency for allrelevant times, this causes errors in the digital signals output to eacharray element transducer. Thus, it is necessary for the development of asignal generation system that is capable of producing a digital signalusing the free selection of amplitude and phase. Thus is used to producea substantially error-free signal that preserves the amplitude and phaserelative to a constant base frequency while allowing the carrierfrequency to vary.

SUMMARY

The solution presented herein uses the specified amplitude and phase toreconstruct a digital signal that explicitly induces the frequencyshifts away from the base frequency implied by phase changes. Shiftingthe carrier frequency of a digitally controlled phased array whilepreserving the timing of the individual phase pulses enables moreefficient driving of the phased array system when the phase of the drivesignals change dynamically in time.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying figures, where like reference numerals refer toidentical or functionally similar elements throughout the separateviews, together with the detailed description below, are incorporated inand form part of the specification, serve to further illustrateembodiments of concepts that include the claimed invention and explainvarious principles and advantages of those embodiments.

FIG. 1 shows an output from a quickly-moving focus point using astandard approach.

FIG. 2 shows an output similar to FIG. 1 but from a quickly-moving focuspoint using a novel approach.

FIGS. 3A and 3B are closeups of FIGS. 1 and 2, respectively.

FIG. 4 shows geometric behavior of a system having 3/2 of a basefrequency.

FIG. 5 shows geometric behavior of a system having exactly the basefrequency.

FIG. 6 shows geometric behavior of a system having a slowly increasingphase relative to the base frequency.

FIG. 7 shows geometric behavior of a system having an arbitrary functionof phase angle relative to the base frequency.

FIG. 8 shows geometric construction of proof of the correct behavior forthe case of increased frequency.

FIG. 9 shows geometric construction of proof of the correct behavior forthe case of decreased frequency.

Skilled artisans will appreciate that elements in the figures areillustrated for simplicity and clarity and have not necessarily beendrawn to scale. For example, the dimensions of some of the elements inthe figures may be exaggerated relative to other elements to help toimprove understanding of embodiments of the present invention.

The apparatus and method components have been represented whereappropriate by conventional symbols in the drawings, showing only thosespecific details that are pertinent to understanding the embodiments ofthe present invention so as not to obscure the disclosure with detailsthat will be readily apparent to those of ordinary skill in the arthaving the benefit of the description herein.

DETAILED DESCRIPTION

Carrier frequency is defined herein as the consequential instantaneousfrequency of the output digital signal pulses. Base frequency is definedherein as the center frequency that is described by an unmoving orconstant phase signal. It is known that any phase change in the signalalso constitutes a frequency shift. In this case that is realized by thecarrier frequency shifting away from the base frequency. To preservecompatibility with the input intended for the elements of a phasedarray, the input data to the method is considered to be the phase andduty cycle of a pulse-width modulation. This is measured with respect toa steady reference signal that is a fixed source at the base frequency.

I. Shifting Carrier Frequencies While Preserving Time and Phase Accuracy

Prior solutions suppose a set carrier frequency (that is always equal tothe base frequency) where any variation from this frequency (includingany change in phase) generates error. If the true carrier frequency isclose to the base frequency this error is small, but it is stillconsistently present when (for instance) moving the focus of the phasedarray.

FIG. 1 shows an output from a quickly-moving focus point using astandard approach. The drive of each transducer in a square 16×16element phased array is represented by a column of black and white data.Horizontal gray bars represent the start of each period of the basefrequency and the rows are each tick of the clock on which a binarydigit that describes the transducer state must be output. Each square ofblack or white data describes the binary state of the electrical inputsignal and thus transducer element at that tick through time. Eachcarrier frequency period is generated independently at the basefrequency with the correct phase and duty cycle that corresponds to theamplitude.

Specifically, FIG. 1 is a visualization 100 of the digital pulse-widthmodulated output from the standard technique when the focus for thearray is moving quickly. The input to each transducer is arrangedhorizontally 160, with time 150 moving from the bottom to the top of thediagram. The base frequency periods are delineated 161, 162, 163, 164,showing where one period ends and the next begins. Erroneously repeatingpulses 110, 120, 130, 140 can be seen that are indicative of a frequencyshift from the base frequency, here to a lower carrier frequency. Thestandard technique requires that each base frequency period contains atleast one pulse region 110, 120, 130, 140. But it can be seen that thecorrect behavior should involve fewer than one pulse region per basefrequency period, showing that the system is attempting to generate afrequency lower than the base frequency. This can be appreciated by thecurving trend of the pulses required to affect a focus in the phasedarray. The trend of these pulses formed by the focusing process haveerroneous discontinuities at the edges of each period. This trend can beidentified as seeming to have repeated discontinuous offsets in time inorder to fulfil the implied requirement of having one pulse region perbase frequency period.

The new algorithm, whose output is shown in FIG. 2, allows the carrierfrequency to shift away from the base frequency while preserving thetiming and phase accuracy of the amplitude generating pulses. Shown is avisualization 200 of the digital pulse-width modulated output from thenew technique when the focus fir the array is moving quickly. The inputto each transducer is arranged horizontally 260, with time 250 movingfrom the bottom of the top of the diagram. The base frequency periodsare delineated 261, 262, 263, 264, showing where one period ends and thenext begins. Here the double pulses 210, 220, 230, 240 are no longer aproblem. While the phases agree exactly with the standard approach atthe start of each base frequency period, the number of pulse regions perbase frequency period is now on average less than one. These featurestaken together show that the carrier frequency has been loweredcorrectly.

The detailed differences between the output from both approaches isshown in FIGS. 3A and 3B. FIG. 3A is a detail 300 of the upper leftcorner of FIG. 1. FIG. 3B is a detail 350 of the upper left corner ofFIG. 2. As the waveform of both techniques are equivalent at the startof each base frequency period 162, 262 the pulse intended to be producedat the base frequency in the old technique (FIG. 3A), has insteadproduced a double pulse 310 due to the moving phase shift. The realityis that not only is the double pulse incorrect, its frequency content isdisruptive to the behavior of the system. This is because the temporaldistance between the two pulses implies very high frequency content thatis near neither the base nor the intended carrier frequency.

In the new technique (FIG. 3B), a single pulse 360 has been produced ata lower frequency that corresponds to the corrected position across bothbase frequency periods (between 261 and 262 and between 262 and 263).The single pulse 360 further has been elongated in time to factor in thechange in the number of discrete ticks that a given duty cyclepercentage implies. This is shown by its occupation of more pulseelements (white squares) than the equivalent pulses in 310.

II. Hardware Techniques to Create Suitable PWM Output

The hardware-efficient method of generating the up-sampling of the phaserepresented as the evaluation of high-order polynomial interpolant isalso novel. The aim is to produce a PWM output that respects andcorrectly interprets changes in frequency while also preserving absolutephase and phase changes. Without loss of generality, this technique maybe also restated with phase delays that produces a “sign flip” in anglefrom the technique described.

The moving phase angle may be considered as an equivalent formulationsfor a phase-frequency modulated wave:

${{\cos\left( {{\omega t} + {\int{\frac{d{\theta^{\prime}(t)}}{dt}{dt}}} + \theta} \right)} = {{\cos\left( {{\theta^{\prime}(t)} + {\omega t}} \right)} = {\cos\left( {\theta + {\int{{\omega^{\prime}(t)}{dt}}}} \right)}}},$

where θ′(t) is a time-dependent function of phase and ω′(t) is atime-dependent function of frequency. It can be seen that dθ′(t)/dt is ameasure of deviation of the carrier frequency from the base frequency.This can be simplified by normalizing both angle and ω (divide throughby the base frequency and 2π radians, θ now being measured inrevolutions), yielding ω=1.

Describing the phase delay θ′(t) may be achieved by interpolating phaseoffsets generated in subsequent base frequency steps by a polynomial,since it is beneficial for the frequency to be defined and continuous onthe endpoints. The frequency is defined as:

${{\frac{d{\theta^{\prime}(t)}}{dt} + \omega} = {\omega^{\prime}(t)}},$

where the first time derivatives of phase angle also contribute to theinstantaneous carrier frequency and thus form two derivativeconstraints:

${{\frac{d{\theta^{\prime}(0)}}{dt} + \omega} = {{\omega^{\prime}(0)} = {\omega + \theta_{0} - \theta_{- 1}}}},{{\frac{d{\theta^{\prime}(1)}}{dt} + \omega} = {{\omega^{\prime}(1)} = {\omega + \theta_{1} - {\theta_{0}.}}}}$

The two endpoints of the interval in angle also have furtherconstraints:

θ′(0)=θ_(0,)

θ′(1)=θ_(1,)

which together with the constraints on carrier frequency make four intotal. This necessitates a cubic polynomial interpolation for this levelof continuity. As shown, defining ω′(0) and ω′(1) can be achieved usingbackwards differences, thus limiting the number of samples required inthe future direction and reducing latency. This also reduces the totalnumber of immediately available samples required from four to threeprecomputed samples of the phase angle and duty cycle of the intendedsignal.

The cubic form of the interpolating spline polynomial formed frombackwards differences is:

θ′(t)=(−θ⁻¹+2θ₀−θ₁)t ³+(2θ⁻¹−4θ₀+2θ₁)t ²+(θ₀−θ⁻¹)t+θ ₀,

which is repeated for every interval.

Further, the phase may be also represented by a lower degree polynomial.Although this would imply sacrificing some of the continuity conditions,the reasonable approach is to produce discontinuities in frequencyanyway (but importantly, phase continuity is preserved as only the timederivatives of the phase are discontinuous). Even with frequencydiscontinuities, the technique using this interpolant enjoys asignificant accuracy improvement over the standard technique. The linearinterpolant for such a method may be stated as:

θ′(t)=(θ₁−θ₀)t+θ ₀.

Although the complexity of the implementation increases, higher orderinterpolation polynomials may equally be used without loss ofgenerality. The on-time of a digital signal is described by the dutycycle, which is assumed proportional to the amplitude of the signal.This motivates the name “pulse-width modulation.” This can be realizedhere by adding an interpolation on the duty cycle value Δ of the signalencoded as a pulse-width percentage at the base frequency:

Δ′(t)=(Δ₁−Δ₀)t+Δ ₀.

Defining the output signal going into the element as a digitalapproximation to:

${{\cos\left( {{\omega t} + {\int{\frac{d{\theta^{\prime}(t)}}{dt}{dt}}} + \theta} \right)} = {\cos\left( {{\theta^{\prime}(t)} + {\omega t}} \right)}},$

so a time-varying θ phase offset with respect to the base frequency mayalso be viewed as a deviation from the base signal frequency ω,effectively dθ′(t)/dt. To search for the locations of the pulses, zeroes(also multiples of 2π) of the angle input to the cosine function must befound. These correspond to peaks in the wave and high points in thedigital signal. To achieve this, both angle and ω are normalized (dividethrough by the frequency and 2π radians, all θ now being measured inrevolutions), yielding ω=1. Therefore, the condition being searched foris:

ωt−θ′(t)=t−θ′(t)=0.

This describes the center of the pulse at each step.

To find the extent of the pulse around the center point, the value|t−θ′(t)| is computed. If it is smaller than a given value representingan amplitude, then the point in time is within the pulse, in the highregion of the digital signal. Otherwise, the point in time is outsidethe pulse and in the low region of the digital signal.

FIGS. 4 through 7 geometrically demonstrate how testing that this valueis less than Δ′(t)/2 generates the appropriate pulse.

FIG. 4 shows the geometrical behavior 400 for the edge case of 3/2 ofthe base frequency. In this graph, Δ is Δ(t), the y-axis 410 representsnormalized angle (in revolutions) θ, and the x-axis 420 representsnormalized time (in base frequency periods) t. This FIG. 4 is ageometric interpretation of the PWM generation when applied to a slowlydecreasing phase (with derivative −½, negative slope down and to theright), relative to the base frequency represented by the diagonal phaselines 440 a, 440 b, 440 c.

The distance between the two sets of repeating curves crosses thethreshold where it is less than Δ/2 (defined as half the duty cyclequantity) distance in a number of places 450 a, 450 b, 450 c, 450 d, 450e, 450 f, 450 g, 450 h that repeat in time. These two sets of curves arethe constant phase versus timelines 460 a, 460 b, 460 c, 460 d, 460 e(θ=t or θ=ωt, but wrapped around in rotations and base frequency periodssince ω is normalized to one). This travels up and to the right of thediagram that represent the base frequency with zero phase offsetbehavior. The interpolated phase curves (θ′(t)) that represent thedesired behavior that are an addition to this signal in phase 440 a, 440b, 440 c. Where the two curves “match” in phase closely enough (lessthan Δ/2), these regions represent the pulse parts of the pulse signals430. The dashed vertical lines projected from the Δ/2 distance factors450 a, 450 b, 450 c, 450 d, 450 e, 450 f, 450 g, 450 h show the placeson the PWM signal 430 where the binary state is changed inducing pulseedges due to the Δ/2 distance factor being reached.

The constant phase versus timelines (θ=t or θ=ωt) travelling up and tothe right of the diagram that represent the base frequency with zerophase offset behavior are repeated for every period of the basefrequency. The repetition in the vertical direction shows that it istrue for all integer numbers of rotations in angle. Thus it is true evenconsidering numerical wrap-around of the counters used to implement themethod. This generates a PWM signal with a carrier frequency that isthree-halves the base frequency (where the frequency multiplier isobtained by subtracting the instantaneous derivative of the interpolatedphase lines θ′(t) (−½) from the derivative of the constant phase versustimelines θ=ωt (1), so 1−(−½)= 3/2). At the bottom is the final digitalsignal 430 that is to drive the element made up of all of the pointswhere the two sets of curves are less than Δ/2 distance apart.

FIG. 5 shows the geometrical behavior 500 for exactly the basefrequency. In this graph. Δ is Δ(t), the y-axis 510 representsnormalized angle (in revolutions) θ, and the x-axis 520 representsnormalized time (in base frequency periods) t. Shown is a geometricinterpretation of the PWM generation when applied to a flat constantphase angle θ′(t) (horizontal lines with derivative zero in time) thatdoes not change relative to the base frequency represented by thediagonal phase lines 540 a, 540 b, 540 c.

The distance Δ/2 550 a, 550 b, 550 c, 550 d, 550 e, 550 f againrepresents the transition points between the two states in the pulsesignal. Thus, the two curves cross over exactly once per base frequencyperiod because the interpolated phase curve is horizontal and representsa constant phase angle. This generates a PWM signal with a carrierfrequency that that is exactly equal to the base frequency (where thefrequency multiplier is again obtained by subtracting the instantaneousderivative of the interpolated phase lines θ′(t)(θ) from the derivativeof the constant phase versus timelines θ=ωt (1), so 1−0=1). The dashedlines show the pulse edges in the pulsed signal. At the bottom is thefinal digital signal 530 that is to drive the element made up of all ofthe points where the two sets of curves are again less than Δ/2 distanceapart.

FIG. 6 shows the geometrical behavior 600 for the edge case of ½ thebase frequency. In this graph, Δ is Δ(t), the y-axis 610 representsnormalized angle (in revolutions) θ, and the x-axis 620 representsnormalized time (in base frequency periods) t. The distance Δ/2 650 a,650 b again represents the transition points between the two states inthe pulse signal.

Shown is a geometric interpretation of the PWM generation when appliedto an increasing phase θ′(t) (with derivative ½), relative to the basefrequency represented by the diagonal phase lines 640 a, 640 b, 640 c.This generates a PWM signal with a carrier frequency that is half thebase frequency (where the frequency multiplier is obtained bysubtracting the instantaneous derivative of the interpolated phase linesθ′(t)(+½) from the derivative of the constant phase versus time linesθ=ωt (1), so 1−(+½)=½). At the bottom is the final digital signal 630that is to drive the element made up of all of the points where the twosets of curves are again less than Δ/2 distance apart.

FIG. 7 shows the geometrical behavior 700 as to how an exampleinterpolated function where the gradient and thus frequency changessignificantly over time fits into this geometric description. In thisgraph, Δ is Δ(t), the y-axis 710 represents normalized angle (inrevolutions) θ, and the x-axis 720 represents normalized time (in basefrequency periods) t. Shown is a geometric interpretation of the PWMgeneration when applied to a more arbitrarily defined function of phaseangle, relative to the base frequency represented by the diagonal phaselines θ′(t) or θ=ωt 740 a, 740 b, 740 c, 740 d, 740 e, 740 f, 740 g, 740h, 740 j, 740 k. The distance Δ/2 750 a, 750 h, 750 c, 750 d, 750 e, 750f, 750 g, 750 h, 750 j, 750 k, 750 m, 750 n, 750 p, 750 q, 750 r, 750 s,750 t, 750 u, 750 v, 750 w again represents the edges in the pulsesignal. But here while they represent the same Δ/2 distance on they-axis, they correspond to varying pulse length on the x-axis. The wavyhorizontal lines 745 a, 745 b, 745 c are the interpolated phase linesθ′(t) in this example.

The variation in the derivative of θ′(t) moves between positivederivative that generates longer pulses at a lower frequency andnegative derivative that generates shorter pulses at a higher frequency.This is due to the crossings between y-axis distances smaller than Δ/2and larger than Δ/2 changing their relative distance apart. At thebottom is the final digital signal 730 that is to drive the element,wherein pulse edges are induced when the signal y-axis distance crossesthe Δ/2 threshold.

It can also be proven that the duty cycle value Δ′(t)/2 when used inthis way scales appropriately with frequency for this scheme.

FIG. 8 shows the geometric construction 800 of the proof of the correctbehavior for the case of increased frequency. The y-axis 810 representsnormalized angle (in revolutions) θ and the x-axis 820 representsnormalized time (in base frequency periods) t. The distance Δ/2 844again represents the state change in the transducer drive signal. But inthis situation the curve θ′(t) is assumed to have negative derivative

$\frac{d{\theta^{\prime}(t)}}{dt},$

resulting in a higher frequency. The more negative the derivative

$\frac{d{\theta^{\prime}(t)}}{dt},$

the higher the frequency and the larger the distance X should be.

Further, FIG. 8 shows a geometric proof of the correct scaling of theamplitude of the pulses with frequency. This is a further effect of themeasurement approach for carrier frequencies that have increased withrespect to the base frequency. This is demonstrated by showing that p/2836, 838, which is half the width of the final pulse, scales in timeappropriately with the frequency shift. Further, a negatively slopedphase curve θ′(t) 834 that is of constant slope intersects 830 apositively sloped curve 832 θ=t. To find p/2 836, 838 (the length ofhalf of the pulse at the new higher frequency) the angle α 840 can befound as:

${{\tan\alpha} = {\frac{x}{p/2} = {❘\frac{d{\theta^{\prime}(t)}}{dt}❘}}},$

and since

${\frac{\Delta^{\prime}(t)}{2} = {\frac{p}{2} + x}},{x = {\frac{p}{2}{❘\frac{d{\theta^{\prime}(t)}}{dt}❘}}},{\frac{\Delta^{\prime}(t)}{2} = {\frac{p}{2} + {\frac{p}{2}{❘\frac{d{\theta^{\prime}(t)}}{dt}❘}}}},{\frac{\Delta^{\prime}(t)}{2} = {\frac{p}{2}\left( {1 + {❘\frac{d{\theta^{\prime}(t)}}{dt}❘}} \right)}},{\frac{p}{2} = \frac{{\Delta^{\prime}(t)}/2}{1 + {❘\frac{d{\theta^{\prime}(t)}}{dt}❘}}},$

which is the definition of the appropriate pulse-width change due to thefrequency shift when

$\frac{d{\theta^{\prime}(t)}}{dt} \leq 0.$

In summary, FIG. 8 demonstrates that the half pulse width isappropriately scaled in this situation for the frequency multiplier

${1 + {❘\frac{d{\theta^{\prime}(t)}}{dt}❘}},$

when the y-axis distance Δ/2 is used as the criterion for the pulseedges. This is exactly the frequency multiplier required.

FIG. 9 shows the geometric construction 900 of the proof of the correctbehavior for the case of decreased frequency. The y-axis 910 representsnormalized angle (in revolutions) θ and the x-axis 920 representsnormalized time (in base frequency periods) t. The distance Δ/2 944again represents the state change in the transducer drive signal. But inthis situation the curve θ′(t) is assumed to have positive derivative

$\frac{d{\theta^{\prime}(t)}}{dt},$

resulting in a lower frequency. The more positive the derivative

$\frac{d{\theta^{\prime}(t)}}{dt},$

the lower the frequency and the larger the distance X should be.

Further, shown is a geometric proof of the correct scaling of theamplitude of the pulses with frequency. This is a further effect of themeasurement approach for carrier frequencies that have increased withrespect to the base frequency. This is demonstrated by showing that p/2936, 942 which is half the width of the final pulse, scalesappropriately in time with the frequency shift.

Further, a positively sloped phase curve θ′(t) 934 that is of constantslope, intersects 930 a positively sloped curve θ=t 932. To find p/2936, 942 (the length of half of the pulse at the new lower frequency)the angle α 940 can be found using a similar construction as before as:

${{\tan\alpha} = {\frac{x}{p/2} = {❘\frac{d{\theta^{\prime}(t)}}{dt}❘}}},$

and since

${\frac{\Delta^{\prime}(t)}{2} = {\frac{p}{2} - x}},{x = {\frac{p}{2}{❘\frac{d{\theta^{\prime}(t)}}{dt}❘}}},{\frac{\Delta^{\prime}(t)}{2} = {\frac{p}{2} - {\frac{p}{2}{❘\frac{d{\theta^{\prime}(t)}}{dt}❘}}}},{\frac{\Delta^{\prime}(t)}{2} = {\frac{p}{2}\left( {1 - {❘\frac{d{\theta^{\prime}(t)}}{dt}❘}} \right)}},{\frac{p}{2} = \frac{{\Delta^{\prime}(t)}/2}{1 - {❘\frac{d{\theta^{\prime}(t)}}{dt}❘}}},$

which is the definition of the appropriate pulse width change due to thefrequency shift when

$\frac{d{\theta^{\prime}(t)}}{dt} \geq 0.$

In summary, FIG. 9 demonstrates that the half pulse width isappropriately scaled in this situation for the frequency multiplier

${1 - {❘\frac{d{\theta^{\prime}(t)}}{dt}❘}},$

when the y-axis distance Δ/2 is used as the criterion for the pulseedges. This is exactly the frequency multiplier required.

In conclusion, having shown that the Boolean test:

|t−θ′(t)|<Δ′(t)/2,

is guaranteed to produce the best approximation of the pulse, it isimperative that the progressive evaluation of the polynomial interpolantbe achieved with an efficient hardware algorithm.

III. Counter Architecture

The uniformly progressive requirement for the evaluation of the splineinterpolant affords an intuitive and low-cost approach to the evaluationof polynomial. By incrementing time forward, a series of cascadedcounters may be used to derive simplex numbers and thus the powers of trequired, For each carrier frequency period, t should range in theinterval 0 to 1 to evaluate the spline polynomial.

Each spline interval counts k discrete sub-interval steps from 0 to2^(INT_BITS)−1. By incrementing each counter, it can compute the termscorresponding to the powers of k. By reinterpreting powers of k asfractions, powers of t are computed. This occurs by first establishing aper spline-curve constant, for instance taking the constant b. Then, thefirst counter (proportional to b multiplied by a linear order term in t)is defined by:

∝bO(t)₀=0,

∝bO(t)_(k) =∝bO(t)_(k−1) +b,

∝bO(t)_(k) =bk=2^(INT_BITS) bt.

From this counter, it is clear that bit shifting the sum by INT_BITSimmediately yields the first power oft multiplied by the increment bt.It should be noted that if only linear interpolants are required, theymay be calculated through the application of this counter wheninitializing to the constant part of the interpolant. For instance, ifbt+ƒ were required:

∝(ƒ+bO(t))₀=2^(INT) ^(BITS) ƒ,

∝(ƒ+bO(t))_(k)=∝(ƒ+bO(t))_(k−1) +b,

∝(ƒ+bO(t))_(k)=2^(INT_BITS) ƒ+bk=2^(INT_BITS)(bt+ƒ).

Continuing with the high-order interpolant, the next counter(proportional to b multiplied by a quadratic order term in t) is definedby the “triangular” numbers as:

${{\propto {b{O\left( t^{2} \right)}_{0}}} = 0},{{\propto {b{O\left( t^{2} \right)}_{k}}} = {\propto {{b{O\left( t^{2} \right)}_{k - 1}} +} \propto {b{O(t)}_{k}}}},{{\propto {b{O\left( t^{2} \right)}_{k}}} = {{b\left( {{\frac{1}{2}k^{2}} + {\frac{1}{2}k}} \right)} = {b\frac{k\left( {k + 1} \right)}{2!}}}},{{{2\left( {\propto {b{O\left( t^{2} \right)}_{k}}} \right)} - \left( {\propto {b{O(t)}_{k}}} \right)} = {2^{2 \times {INT}_{BITS}}{{bt}^{2}.}}}$

The linear part must be subtracted beforehand due to the difference inthe power-of-two. This then yields the quadratic term in t, bt², whenadjusted to represent a fraction.

The next counter (and the final one required for a cubic interpolant,which is proportional to b multiplied by a cubic order term in t) isdefined by the “tetrahedral” numbers as:

${{\propto {b{O\left( t^{3} \right)}_{0}}} = 0},{{\propto {b{O\left( t^{3} \right)}_{k}}} = {\propto {{b{O\left( t^{3} \right)}_{k - 1}} +} \propto {b{O\left( t^{2} \right)}_{k}}}},{{\propto {b{O\left( t^{3} \right)}_{k}}} = {{b\left( {{\frac{1}{6}k^{3}} + {\frac{1}{2}k^{2}} + {\frac{1}{3}k}} \right)} = {b\frac{{k\left( {k + 1} \right)}\left( {k + 2} \right)}{3!}}}},{{{{6\left( {\propto {b{O\left( t^{3} \right)}_{k}}} \right)} - \left( {\propto {b{O\left( t^{2} \right)}_{k}}} \right) +} \propto {b{O(t)}}} = {2^{3 \times {{INT}\_{BITS}}}{bt}^{3}}},$

In some embodiments in the expressions for higher powers of t, due tothe differences in the power-of-two prefixed it may be possible to omitthe corrections when the power oft of the error term and the power of tof the desired term are separated by some c orders of magnitude. Thissituation may incur acceptable levels of error for sufficiently large c.This can instead be described using the k−1^(th) iteration to generate:

${{\propto {b{O\left( t^{3} \right)}_{- 1}}} = 0},{{\propto {b{O\left( t^{3} \right)}_{k - 1}}} = {{b\left( {{\frac{1}{6}k^{3}} - {\frac{1}{6}k}} \right)} = {b\frac{\left( {k - 1} \right){k\left( {k + 1} \right)}}{3!}}}},{{{{6\left( {\propto {b{O\left( t^{3} \right)}_{k - 1}}} \right)} +} \propto {b{O(t)}}} = {2^{3 \times {{INT}\_{BITS}}}{bt}^{3}}},$

By combining these powers of t, potentially with different substituted bdepending on the construction of the spline polynomial, simple countersmay be used to generate splines consisting of arbitrarily high-orderpolynomials. The incremented counters serve to increment the timevariable t forward, and in so doing, generate the next discrete timeslice of the intended signal.

For instance, given the interpolating spline:

θ′(t)=(−θ⁻¹+2θ₀−θ₁)t ³+(2θ⁻¹−4θ₀+2θ₁)t ²+(θ₀−θ⁻¹)t+θ ₀,

and the duty cycle interpolant:

Δ′(t)=(Δ₁−Δ₀)t+Δ ₀,

the counters required are then:

A _(counter):=∝(Δ₀+(Δ₁−Δ₀)O(t)),

B _(counter):=∝((θ⁻¹−2θ₀+θ+θ₁)O(t)),

C _(counter):=∝((θ⁻¹−2θ₀+θ+θ₁)O(t ²)),

D _(counter):=∝((θ⁻¹−2θ₀+θ₁)O(t ³)),

E _(counter):=∝(θ₀+(θ₀−θ⁻¹)O(t)).

To assemble Δ′(t) (the duty cycle to achieve the required output powerat the base frequency) and θ′(t) (the instantaneous phase offset withrespect to the base frequency) these may be combined as:

${{\Delta^{\prime}(t)} = {{\Delta_{0} + {\left( {\Delta_{1} - \Delta_{0}} \right)t}} = \frac{A_{counter}}{2^{{INT}\_{BITS}}}}},{{{\left( {\theta_{0} - \theta_{- 1}} \right)t} + \theta_{0}} = \frac{E_{counter}}{2^{{INT}_{BITS}}}},{{\left( {{2\theta_{- 1}} - {4\theta_{0}} + {2\theta_{1}}} \right)t^{2}} = \frac{{4C_{counter}} - {2B_{counter}}}{2^{2 \times {INT}_{BITS}}}},{{\left( {{- \theta_{- 1}} + {2\theta_{0}} - \theta_{1}} \right)t^{3}} \approx \frac{6D_{counter}}{2^{3 \times {INT}_{BITS}}}},$

which are trivially added together and subtracted to produce the Booleancondition that drives the output signal.

IV. Low Bandwidth Operation Versus Large Frequency Bandwidth

It should also be noted that due to the power-of-two behaviors of thecounter architecture, this may be applied to phase and amplitude samplesspaced by power-of-two counts of the base frequency period and not justsplitting the base frequency period into 2^(INT_BITS) discrete steps.This could be used in some embodiments to provide a lower bandwidthinterface, both in data usage and in frequency range about the basefrequency that are addressable. In a similar vein, the phase andamplitude samples may be increased in some embodiments to have apower-of-two samples in every base frequency period to provideadditional frequency bandwidth about the carrier instead.

To achieve lower data rate at the cost of reduced frequency bandwidthabout the base frequency, the spline interpolant interval must be madeto cover a power-of-two count of base frequency periods. This mayinvolve increasing the number of bits of precision produced by theinterpolant or decreasing the number of bits of precision of the timercounter, so that 2^(INT_BITS) spans multiple periods. Each areaddressable by the most significant bits of INT_BITS. Further, multipleprogressions of the wrapped-around θ=t line must be supported in thespline interval in this mode of operation, making the de facto splineinterval 0≤t<2^(N). This decreases the accessible range of frequencies,since the maximum angular movement of the additional phase, π radians,is now spread across 2^(N) base frequency periods. This reduces themaximum possible frequency shift.

To achieve increased frequency bandwidth about the base frequency at theexpense of increased data rate, the spline interpolant interval must bereduced to cover only a power-of-two with a negative exponent(fractional) part of the base frequency period. This may involvedecreasing the number of bits of precision produced by the interpolantor increasing the number of bits of precision of the timer counter. Theaddition of these extra data points effectively makes the splineinterval 0≤t<2^(−N) This increases the accessible range of frequenciesas the maximum angular movement of the additional phase, π radians, isnow confined to 2^(−N) fraction of base frequency period. This increasesthe maximum possible frequency shift achievable.

V. Environmental Compensation

With traditional techniques such as frequency multipliers, dividers andphased locked loops (PLLs), it is difficult to make small or dynamicchanges to the carrier frequency. As the dynamic control of a phasedarray requires exact phase information this implies that a digitalapproach, if the hardware is sufficiently capable, will be more suitablethan an analog solution for prescribing precise behavior. Thus, itsbehavior is deterministic, easy to model and predictable compared to asystem which uses analog infrastructure to achieve this level ofsynchronization due to delays and instabilities among other problems. Asthe system explicitly evaluates the phase function, the transducerimpulse response can be predicted and taken into account by modeling thefrequency content of the output signal. Further, with the techniquedescribed herein, by adding an accumulator onto the phase angle inputinto the signal generator described produces a frequency shift that maybe easily controlled without affecting the relative phases in an unknownway. In the case of environmental changes, it may be that the frequencyrequires such small changes to keep the wavelength constant. Theremainder of the system may then continue to work in units of wavelengthwith the guarantee that this is compensated for by the driving system atthe end of the pipeline. The accumulator that adds onto the phase forall of the transducers is incremented or decremented using a value thatis derived from one or more sensors, such as (for example) temperature,humidity and altitude/density sensors.

VI. Further Description

The following paragraphs provide further description of the invention.

1. A system comprising a digital or analog electrical signal driven by areal-time progressive polynomial spline evaluation that achieves anup-sampling by a power-of-two factor using a linear combination ofcounters.

2. A system comprising a digital or analog electrical signal whoseinstantaneous phase angle is substantially calculated by a real-timeprogressive polynomial spline evaluation that achieves an up-sampling bya power-of-two factor using a linear combination of counters.

3. A system comprising a digital electrical signal whose state iscalculated by comparing the difference between the current position inthe base frequency cycle and the instantaneous phase angle to theproportion of cycle duty that would be present at the base frequency.

4. The system of claim 1, 2 or 3 where the polynomial spline interval isa base frequency period.

5. The system of claim 1, 2 or 3 where the polynomial spline interval isa power-of-two count of base frequency periods.

6. The system of claim 1, 2 or 3 where the polynomial spline interval isa power-of-two fraction of the base frequency period.

7. The system of claims 1-6 where the phase angle is modified by arolling counter incremented by a value driven by external sensorsmonitoring the environment.

8. The system of claims 1-6 where the phase angle is modified by arolling counter decremented by a value driven by external sensorsmonitoring the environment.

VII. Conclusion

While the foregoing descriptions disclose specific values, any otherspecific values may be used to achieve similar results. Further, thevarious features of the foregoing embodiments may be selected andcombined to produce numerous variations of improved haptic systems.

In the foregoing specification, specific embodiments have beendescribed. However, one of ordinary skill in the art appreciates thatvarious modifications and changes can be made without departing from thescope of the invention as set forth in the claims below. Accordingly,the specification and figures are to be regarded in an illustrativerather than a restrictive sense, and all such modifications are intendedto be included within the scope of present teachings.

Moreover, in this document, relational terms such as first and second,top and bottom, and the like may be used solely to distinguish oneentity or action from another entity or action without necessarilyrequiring or implying any actual such relationship or order between suchentities or actions. The terms “comprises,” “comprising,” “has”,“having,” “includes”, “including,” “contains”, “containing” or any othervariation thereof, are intended to cover a non-exclusive inclusion, suchthat a process, method, article, or apparatus that comprises, has,includes, contains a list of elements does not include only thoseelements but may include other elements not expressly listed or inherentto such process, method, article, or apparatus. An element proceeded by“comprises . . . a”, “has . . . a”, “includes . . . a”, “contains . . .a” does not, without more constraints, preclude the existence ofadditional identical elements in the process, method, article, orapparatus that comprises, has, includes, contains the element. The terms“a” and “an” are defined as one or more unless explicitly statedotherwise herein. The terms “substantially”, “essentially”,“approximately”, “about” or any other version thereof, are defined asbeing close to as understood by one of ordinary skill in the art. Theterm “coupled” as used herein is defined as connected, although notnecessarily directly and not necessarily mechanically. A device orstructure that is “configured” in a certain way is configured in atleast that way but may also be configured in ways that are not listed.

The Abstract of the Disclosure is provided to allow the reader toquickly ascertain the nature of the technical disclosure. It issubmitted with the understanding that it will not be used to interpretor limit the scope or meaning of the claims. In addition, in theforegoing Detailed Description, various features are grouped together invarious embodiments for the purpose of streamlining the disclosure. Thismethod of disclosure is not to be interpreted as reflecting an intentionthat the claimed embodiments require more features than are expresslyrecited in each claim. Rather, as the following claims reflect,inventive subject matter lies in less than all features of a singledisclosed embodiment. Thus, the following claims are hereby incorporatedinto the Detailed Description, with each claim standing on its own as aseparately claimed subject matter.

1-12. (canceled)
 13. A system comprising: a digital electrical signalhaving a state, wherein the state is calculated by comparing adifference between a current position in a base frequency cycle and aninstantaneous phase angle to a proportion of cycle duty that would bepresent at the base frequency cycle.
 14. The system of claim 13, furthercomprising a polynomial spline having a plurality of intervals, andwherein at least one of the plurality of intervals is a base frequencyperiod.
 15. The system of claim 13, further comprising a polynomialspline having a plurality of intervals, and wherein at least one of theplurality of intervals is a power-of-two count of base frequencyperiods.
 16. The system of claim 13, further comprising a polynomialspline having a plurality of intervals, and wherein at least one of theplurality of intervals is a power-of-two fraction of a base frequencyperiod.
 17. The system of claim 13, wherein the electrical signal has aphase angle, and wherein the phase angle is modified by a rollingcounter incremented by a value driven by an external environmentalmonitoring sensor.
 18. The system of claim 13, wherein the electricalsignal has a phase angle, and wherein the phase angle is modified by arolling counter decremented by a value driven by an externalenvironmental monitoring sensor.